Encyclopedia of Microtonal Music Theory

alpha

First letter of the Greek alphabet, with the following current musical uses of the lower-case form α :

1. Wendy Carlos tuning and scale

A scale devised by Wendy Carlos which does not have octaves, because of its equal step size of ~77.965 cents, or ~15.392 steps per "octave".

It is "defined strictly as a local minimum of the root-mean-square error function that Wendy set up with 3/2, 5/4, 6/5, 7/4, and 11/8 as target intervals." [Paul Hahn, Onelist Tuning Digest 116] ... the plot that Wendy includes with her Computer Music Journal (Spring 1987) article includes three different curves/functions. They include the first three, the first four, and all five of the target intervals listed above. The one from which she derives alpha, beta, and gamma is actually the first, i.e. the one which only targets 3/2, 5/4, and 6/5. [Paul Hahn, Yahoo tuning group, message 2232, Thu Apr 8, 1999 4:35 pm]

The Carlos alpha tuning splits a tempered minor-3rd into two equal parts, which in turn are each divided in half -- thus, the minor-3rd subtends 4 equal steps of ~78 cents each. The alpha tuning also contains a very good approximation to the "pure" pythagorean 3:2 perfect-5th, which is split into 9 equal divisions. The first octave of the alpha scale is covered by the following notes in descending order (step size = 77.965 cents, all figures rounded to the nearest cent):

31-edo is a fairly low-cardinality tuning which gives a rough approximation of the alpha tuning, with 2deg31-edo giving a passable approximation to one step of alpha. But its approximation of the 3/2 ratio is 18deg31-edo, which is the same narrow 5th which acts as the generator in 31-edo's role as the 31-et approximation of 1/4-comma meantone, and the error from the alpha 5th is enough to be audibly noticeable.

77-edo gives a much better approximation of alpha, at the cost of much higher cardinality, with 5deg77-edo closely approximating one step of alpha, and 45deg77-edo giving a very good approximation of the alpha 5th.

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Erv Wilson musical notation

Used in its lower-case form α as an additional letter-name in Erv Wilson's notations, signifying a note near "A", but with a musical function separate from "A", in whatever particular scale the notation is describing.